The course illustrates methods for the theoretical and numerical analysis of nonlinear dynamical systems and their application to case studies in the field of automation engineering. The focus is on the use of nonlinear systems theory, in particular bifurcation analysis, to classify the behaviors of the system and to understand the critical transitions occurring when some model or control parameters are varied. To this aim, the main bifurcations and catastrophes occurring in nonlinear systems are discussed in detail. They are first illustrated on simple examples and then, with the help of tools for numerical analysis, on case studies. A few bifurcations occurring in non-smooth systems (e.g., systems with impacts or switches) are also considered.

The students learn:
• knowledge on (understanding of) the formulation of a nonlinear dynamical system, in continuous and discrete time, as a model of a physical (e.g., mechanical, electrical, chemical) or descriptive (environmental, social, economic) reality;
• knowledge on (understanding of) the mathematical results characterizing the system’s solutions;
• knowledge on (understanding of) the mathematical results characterizing the sensitivity of the system’s behavior to model and control parameters;
• ability to apply the mathematical results to simple case studies;
• ability to use the Matlab toolbox Matcont to numerically analyze bifurcations in real case studies.

Introduction to nonlinear dynamics. Asymptotic behavior of dynamical systems: equilibria, cycles, tori, chaos. Multi-stability. Parameterized systems and structural stability. Bifurcations of equilibria and cycles. Robustness analysis. Hysteresis and catastrophes. Transitions from periodic to aperiodic oscillations. Discontinuous (non-smooth) systems: dynamical properties and bifurcations. Introduction to numerical bifurcation analysis and continuation methods.

The course is organized into: lectures, in which the theoretical notions are presented and illustrated with the help of simple examples; practicums, in which the theoretical notions are applied to simple exercises; seminars, two/three case studies of specific interest for automation engineering (possibly given by external experts, e.g. on power systems, vehicle dynamics, and spacecrafts attitude analysis): computer labs, in which the students will be introduced to the Matlab toolbox Matcont and will work in small groups to numerically study a relatively complex bifurcation problem.

All activities will be integral part of the course program.

Students are required to have basic notions of calculus, geometry, linear and matrix algebra, linear differential equations.

Written test in which students will be asked to:
• solve simple numerical problems on the solutions of a given system and their sensitivity to model and control parameters;
• answer theoretical questions on all topics of the course;
• answer descriptive questions on the seminars;
• answer practical questions on the computer labs.

An assigned laboratory project can optionally contribute to the final assessment.